Consider a wage increase from $10 to $20 an hour, holding nonlabor income V constant.
The wage increase rotates the budget line around the endowment point, as illustrated in Figure 2-8 . The rotation of the budget line shifts the opportunity set from FE to GE. It should be obvious that a wage increase does not change the endowment point: the dollar value of the goods that can be consumed when one does not work is the same regardless of whether the wage rate is $10 or $20 an hour.
The two panels presented in Figure 2-8 illustrate the possible effects of a wage increase on hours of work. In Figure 2-8 a, the wage increase shifts the optimal consumption bundle from point P to point R. At the new equilibrium, the individual consumes more leisure (the increase is from 70 to 75 hours), so that hours of work fall from 40 to 35 hours.
Figure 2-8 b, however, illustrates the opposite result. The wage increase again moves the worker to a higher indifference curve and shifts the optimal consumption bundle from point P to point R. This time, however, the wage increase reduces leisure hours (from 70 to 65 hours), so the length of the workweek increases from 40 to 45 hours. It seems, therefore, that we cannot make an unambiguous prediction about an important question without making even more assumptions.
The reason for the ambiguity in the relation between hours of work and the wage rate is of fundamental importance and introduces a set of tools and ideas that play a central role in all of economics. Both panels in Figure 2-8 show that, regardless of what happens to hours of work, a wage increase expands the worker’s opportunity set. Put differently, a worker has more opportunities when she makes $20 an hour than when she makes $10 an hour. We know that an increase in income increases the demand for all normal goods, including leisure. The increase in the wage thus increases the demand for leisure, which reduces hours of work.
FIGURE 2-8 The Effect of a Change in the Wage Rate on Hours of Work
A change in the wage rate rotates the budget line around the endowment point E. A wage increase moves the worker from point P to point R, and can either decrease or increase hours of work.
U0
U1
E P
R Consumption ($)
Hours of Leisure Slope = −$10
Slope = −$20
70
0 75 110
G
F
V
(a)
U0
U1
E P
R Consumption ($)
Hours of Leisure Slope = −$10
Slope = −$20
65
0 70 110
G
F
V
(b)
But this is not all that happens. The wage increase also makes leisure more expensive.
When the worker earns $20 an hour, she gives up $20 every time she decides to take an hour off. As a result, leisure time is a very expensive commodity for high-wage workers and a relatively cheap commodity for low-wage workers. High-wage workers should then have strong incentives to cut back on their consumption of leisure activities. A wage increase thus reduces the demand for leisure and increases hours of work.
This discussion highlights the essential reason for the ambiguity in the relation between hours of work and the wage rate. A high-wage worker wants to enjoy the rewards of her high income, and hence would like to consume more leisure. The same worker, however, finds that leisure is very expensive and that she simply cannot afford to take time off from work.
These two conflicting forces are illustrated in Figure 2-9 a. As before, the initial wage rate is $10 per hour. The worker maximizes her utility by choosing the consumption bundle given by point P, where she is consuming 70 hours of leisure and works 40 hours per week.
Suppose the wage increases to $20. As we have seen, the budget line rotates and the new consumption bundle is given by point R. The worker is now consuming 75 hours of leisure and working 35 hours. As drawn, the person is working fewer hours at the higher wage.
It helps to think of the move from point P to point R as a two-stage move. The two stages correspond exactly to our discussion that the wage increase generates two effects:
It increases the worker’s income and it raises the price of leisure. To isolate the income effect, suppose we draw a budget line that is parallel to the old budget line (so that its slope is also –$10), but tangent to the new indifference curve. This budget line ( DD ) is also illus- trated in Figure 2-9 a, and generates a new tangency point Q.
FIGURE 2-9 Decomposing the Impact of a Wage Change into Income and Substitution Effects
An increase in the wage rate generates both income and substitution effects. The income effect (the move from point P to point Q) reduces hours of work; the substitution effect (the move from Q to R) increases hours of work.
U1
U1
U0
E P
R
Q Q
D
D D D
Consumption ($)
Hours of Leisure 70
0 75 110
G
F
85
(a) Income Effect Dominates (b) Substitution Effect Dominates U0
E P
R Consumption ($)
Hours of Leisure 65
0 70 110
G
F
80
The move from initial position P to final position R can then be decomposed into a first- stage move from P to Q and a second-stage move from Q to R. It is easy to see that the move from point P to point Q is an income effect. In particular, the move from P to Q arises from a change in the worker’s income, holding wages constant. The income effect isolates the change in the consumption bundle induced by the additional income generated by the wage increase.
Because both leisure and goods are normal goods, point Q must lie to the northeast of point P (so that more is consumed of both goods and leisure). The income effect thus increases the demand for leisure (from 70 to 85 hours) and reduces hours of work by 15 hours per week.
The second-stage move from Q to R is called the substitution effect. It illustrates what happens to the worker’s consumption bundle as the wage increases, holding utility constant. By moving along an indifference curve, the worker’s utility or “real income” is held fixed. The substitution effect thus isolates the impact of the increase in the price of leisure on hours of work, holding real income constant.
The move from point Q to point R illustrates a substitution away from leisure time and toward consumption of other goods. In other words, as the wage rises, the worker devotes less time to expensive leisure activities (from 85 to 75 hours) and increases her consump- tion of goods. Through the substitution effect, therefore, the wage increase reduces the demand for leisure and increases hours of work by 10 hours. The substitution effect implies that an increase in the wage rate, holding real income constant, increases hours of work.
As drawn in Figure 2-9 a, the decrease in hours of work generated by the income effect (15 hours) exceeds the increase in hours of work associated with the substitution effect (10 hours). The stronger income effect thus leads to a negative relationship between hours of work and the wage rate. In Figure 2-9 b, the income effect (again the move from point P to point Q ) decreases hours of work by 10 hours, whereas the substitution effect (the move from Q to R ) increases hours of work by 15 hours. Because the substitution effect domi- nates, there is a positive relationship between hours of work and the wage rate.
The reason for the ambiguity in the relationship between hours of work and the wage rate should now be clear. As the wage rises, a worker faces a larger opportunity set and the income effect increases her demand for leisure and decreases labor supply. As the wage rises, however, leisure becomes more expensive and the substitution effect generates incen- tives for that worker to switch away from the consumption of leisure and instead consume more goods. This shift frees up leisure hours and thus increases hours of work.
To summarize the relation between hours of work and the wage rate:
• An increase in the wage rate increases hours of work if the substitution effect dominates
the income effect.
• An increase in the wage rate decreases hours of work if the income effect dominates the
substitution effect.
2-6 To Work or Not to Work?
Our analysis of the relation between nonlabor income, the wage rate, and hours of work assumed that the person worked both before and after the change in nonlabor income or the wage. Hours of work then adjusted to the change in the opportunity set. But what factors motivate a person to enter the labor force in the first place?
To illustrate the nature of the work decision, consider Figure 2-10 . The figure draws the indifference curve that goes through the endowment point E. This indifference curve indi- cates that a person who does not work at all receives U 0 units of utility. The woman, however, can choose to enter the labor market and trade some of her leisure time for earnings that will allow her to buy consumption goods. The decision of whether to work or not boils down to a simple question: Are the “terms of trade”—the rate at which leisure can be traded for additional consumption—sufficiently attractive to bribe her into entering the labor market?
Suppose initially that the person’s wage rate is given by w low so that the woman faces budget line GE in Figure 2-10 . No point on this budget line can give her more utility than U 0 . At this low wage, the person’s opportunities are quite meager. If the worker were to move from the endowment point E to any point on the budget line GE, she would be mov- ing to a lower indifference curve and be worse off. For example, at point X the woman gets only U G utils. At wage w low , therefore, the woman chooses not to work.
In contrast, suppose that the wage rate was given by w high , so that the woman faces budget line HE. It is easy to see that moving to any point on this steeper budget line would increase her utility. At point Y, the woman gets U H utils. At the wage w high , therefore, the woman is better off working.
In sum, Figure 2-10 indicates that the woman does not enter the labor market at low wage rates (such as w low ), but does enter the labor market at high wage rates (such as w high ). As we rotate the budget line from wage w low to wage w high , we will typically encounter a wage rate, call it w苲 , that makes her indifferent between working and not working. We call w苲 the
The implication that our demand for leisure time responds to its price is not very surprising. When the wage rate is high, we will find ways of minimizing the use of our valu- able time, such as contact a ticket broker and pay very high prices for concert and theater tickets, rather than stand in line for hours to buy a ticket at face value. We will often hire a nanny or send our children to day care, rather than withdraw from the labor market. And we will consume many pre-prepared meals and order pizza or take-out Chinese food, rather than engage in lengthy meal preparations.
It turns out that our allocation of time responds to economic incentives even when there are no easy sub- stitutes available, such as when we decide how many hours to sleep. Sleeping takes a bigger chunk of our time than any other activity, including market work. The typi- cal man sleeps 56.0 hours per week, whereas the typi- cal woman sleeps 56.9 hours per week. Although most persons think that how long we sleep is biologically (and
perhaps even culturally) determined, recent research suggests that, to some extent, sleep time also can be viewed as simply another activity that responds to eco- nomic incentives. As long as some minimum biological threshold for the length of a sleeping spell is met, the demand for sleep time seems to respond to changes in the price of time.
In particular, there is a negative correlation between a person’s earnings capacity and the number of hours spent sleeping. More highly educated persons, for example, sleep less—an additional four years of school decreases sleep time by about one hour per week. Simi- larly, a 20 percent wage increase reduces sleep time by 1 percent, or about 34 minutes per week. When the wage is high, therefore, even dreaming of a nice, long vacation in a remote island becomes expensive.
Source: Jeff E. Biddle and Daniel S. Hamermesh, “Sleep and the Allocation of Time,” Journal of Political Economy 98 (October 1990): 922–943.